Evertse, Jan-Hendrik

Mahler's work on the geometry of numbers

Doc. Math. Extra Vol. Mahler Selecta, 29-43 (2019)
DOI: 10.25537/dm.2019.SB-29-43

Summary

Mahler has written many papers on the geometry of numbers. Arguably, his most influential achievements in this area are his compactness theorem for lattices, his work on star bodies and their critical lattices, and his estimates for the successive minima of reciprocal convex bodies and compound convex bodies. We give a, by far not complete, overview of Mahler's work on these topics and their impact.

Mathematics Subject Classification

11-03, 11H06, 11H16

References

  • [M44]. K. Mahler, Neuer Beweis eines Satzes von A. Khintchine, Rec. Math. Moscow, n. Ser. 1 (1936), 961-962.
  • [M56]. K. Mahler, Ein Übertragungsprinzip für lineare Ungleichungen, Časopis Mat. Fysik. Praha 68 (1939), 85-92.
  • [M57]. K. Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Mat. Fysik. Praha 68 (1939), 93-102.
  • [M75]. K. Mahler, Note on lattice points in star domains, J. London Math. Soc. 17 (1942), 130-133.
  • [M76]. K. Mahler, On lattice points in an infinite star domain, J. London Math. Soc. 18 (1943), 233-238.
  • [M83]. K. Mahler, Lattice points in two-dimensional star domains, I, Proc. London Math. Soc. (2) 49 (1946), 128-157.
  • [M84]. K. Mahler, Lattice points in two-dimensional star domains, II, Proc. London Math. Soc. (2) 49 (1946), 158-167.
  • [M85]. K. Mahler, Lattice points in two-dimensional star domains, III, Proc. London Math. Soc. (2) 49 (1946), 168-183.
  • [M87]. K. Mahler, On lattice points in \(n\)-dimensional star bodies, I: Existence theorems, Proc. R. Soc. London, Ser. A 187 (1946), 151-187.
  • [M88]. K. Mahler, Lattice points in \(n\)-dimensional star bodies, II: Reducibility theorems I, II, III, IV, Proc. Akad. Wet. Amsterdam 49 (1946), 331-343, 444-454, 524-532, 622-631.
  • [M126]. K. Mahler, On compound convex bodies, I, Proc. London Math. Soc. (3) 5 (1955), 358-379.
  • 1. J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in \(\mathbf R}^n\), Invent. Math. 88 (1987), no. 2, 319-340.
  • 2. J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin-New York, 1971, Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 99.
  • 3. H. Davenport, On the Product of Three Homogeneous Linear Forms (II), Proc. London Math. Soc. (2) 44 (1938), no. 6, 412-431.
  • 4. F. J. Dyson, On simultaneous Diophantine approximations, Proc. London Math. Soc. (2) 49 (1947), 409-420.
  • 5. J.-H. Evertse and R. G. Ferretti, A further improvement of the quantitative subspace theorem, Ann. of Math. (2) 177 (2013), no. 2, 513-590.
  • 6. G. Faltings and G. Wüstholz, Diophantine approximations on projective spaces, Invent. Math. 116 (1994), no. 1-3, 109-138.
  • 7. P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, second ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987.
  • 8. E. Hlawka, Zur Geometrie der Zahlen, Math. Z. 49 (1943), 285-312.
  • 9. V. Jarní~k, Eine Bemerkung zum Übertragungssatz, Bŭlgar. Akad. Nauk Izv. Mat. Inst. 3 (1959), no. 2, 169-175 (1959).
  • 10. F. John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187-204.
  • 11. R. Kannan and L. Lovász, Covering minima and lattice-point-free convex bodies, Ann. of Math. (2) 128 (1988), no. 3, 577-602.
  • 12. A. Khintchine, Zwei Bemerkungen zu einer Arbeit des Herrn Perron, Math. Z. 22 (1925), no. 1, 274-284.
  • 13. A. Khintchine, Über eine Klasse linearer Diophantischer Approximationen, Rend. Circ. Math. Palermo 50 (1926), 170-195.
  • 14. H. Minkowski, Geometrie der Zahlen, Teubner, Leipzig, 1910.
  • 15. D. Roy, On Schmidt and Summerer parametric geometry of numbers, Ann. of Math. (2) 182 (2015), no. 2, 739-786.
  • 16. L. A. Santaló, An affine invariant for convex bodies of \(n\)-dimensional space, Portugaliae Math. 8 (1949), 155-161.
  • 17. W. M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526-551.
  • 18. W. M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980.
  • 19. W. M. Schmidt and L. Summerer, Parametric geometry of numbers and applications, Acta Arith. 140 (2009), no. 1, 67-91.
  • 20. W. M. Schmidt and L. Summerer, Diophantine approximation and parametric geometry of numbers, Monatsh. Math. 169 (2013), no. 1, 51-104.

Affiliation

Evertse, Jan-Hendrik
Universiteit Leiden, Mathematisch Instituut, Postbus 9512, 2300 RA Leiden, The Netherlands

Downloads